The structure of human knowledge
My evolving understanding of how different fields relate to each other.
One of the best ways to understand something is to learn it from first principles. You start with some axioms—basic premises that you know to be true—and derive everything else from those. This way you figure out why things are the way they are, rather than memorizing rote facts. You learn things rigorously.
When I entered college, that is exactly what I tried to do. I had an obsession with rigor that drove me from an intended major in biology to one in physics and pure math. I imagined that all of human knowledge was constructed like a building: at the foundation were the "formal" sciences like math and computer science, above which were "physical" sciences like physics and chemistry, and above that still were the biological and social sciences.
I took this view of knowledge-as-a-building very seriously. I wanted my structural foundation to be strong, by avoiding shaky assumptions and concepts that I hadn't constructed from scratch, lest I engage in "hand-waving". When we were learning about organisms and cells in biology class, I felt like I needed to understand molecules and atoms first. But in turn, it didn't make sense to talk about molecules and atoms without understanding sets and numbers. I imagined that the tree of human knowledge grew upon roots in math, logic, and epistemology—the bedrock of everything else we knew.
In this post I'll share why my thinking at the time was mistaken. It was some combination of unrecognized reductionism, a misunderstanding of the historical development of different fields, and above all, an obsession with precision at the expense of everything else.
Reductionism is a family of views about the relationship between the parts of something and the whole. One form of reductionism is the view that to explain something, you need to break it down into parts, and only after fully understanding those parts can you understand the whole. (And, in the extreme, it's the view that understanding the whole is nothing but understanding its parts.)
Reductionism was so deeply embedded in my thinking during college that I didn't even recognize it as an assumption. But it is an assumption, and when it comes to understanding the relationship between different fields, it's not a very good one.1 An explanation of something does not always have to be reductive, and breaking down an entity into parts does not always explain the behavior of the whole.
The physicist David Deutsch gives a great example of this when it comes to explaining water boiling in a kettle. Trying to predict when water will boil based on the trajectories of individual molecules is intractable; it's only solvable when we look at higher-level properties like mass and power supply.
If you fill a kettle with water and switch it on, all the supercomputers on Earth working for the age of the universe could not solve the equations that predict what all those water molecules will do – even if we could somehow determine their initial state and that of all the outside influences on them, which is itself an intractable task.
Fortunately, some of that complexity resolves itself into a higher-level simplicity. For example, we can predict with some accuracy how long the water will take to boil. To do so, we need know only a few physical quantities that are quite easy to measure, such as its mass, the power of the heating element, and so on. For greater accuracy we may also need information about subtler properties, such as the number and type of nucleation sites for bubbles. But those are still relatively ‘high-level’ phenomena, composed of intractably large numbers of interacting atomic-level phenomena. Thus there is a class of high-level phenomena – including the liquidity of water and the relationship between containers, heating elements, boiling and bubbles – that can be well explained in terms of each other alone, with no direct reference to anything at the atomic level or below. In other words, the behaviour of that whole class of high-level phenomena is quasi-autonomous – almost self-contained.
To be clear, many explanations in science are reductive, and the prevalence of reductive explanations is probably what led me to assume that that's how all explanations work. But there are also many non-reductive explanations, even some within physics:
Even in physics, some of the most fundamental explanations, and the predictions that they make, are not reductive. For instance, the second law of thermodynamics says that high-level physical processes tend towards ever greater disorder. A scrambled egg never becomes unscrambled by the whisk, and never extracts energy from the pan to propel itself upwards into the shell, which never seamlessly reseals itself. Yet, if you could somehow make a video of the scrambling process with enough resolution to see the individual molecules, and play it backwards, and examine any part of it at that scale, you would see nothing but molecules moving and colliding in strict obedience to the low-level laws of physics. It is not yet known how, or whether, the second law of thermodynamics can be derived from a simple statement about individual atoms.
My college self thought it was impossible to genuinely understand something at the molecular level without understanding everything at the atomic level first. But there is no strict dependency between the higher and lower levels of reality—we can make progress in understanding the higher levels without perfectly describing the lower levels first. Not only that, but per Deutsch's point, it seems that some things may only be explicable when you're looking at the higher levels.
Another way to examine why my thinking was flawed is to consider how knowledge developed over time.
If my earlier view of knowledge were true—if understanding lower-level fields was logically necessary for understanding higher-level fields—then we would have had to discover logic and epistemology and pure math first, and only then could we have moved on to physics, before turning our attention to chemistry and cells and biology. And once we figured out all of those things, we could have finally gotten started with studying individuals and societies.
But of course, the actual development of these fields was much messier than that. Yes, some of the earliest discoveries we made were in math and epistemology, but around the same time we were also making progress in astronomy and medicine and history. We discovered a whole lot of mathematics before we actually established a formal foundation for it in set theory. We were conducting science for a few thousand years even as we were revising our understanding of what "science" actually is.
To this day, from math to epistemology and physics and biology, no one area of study is "solved"—they all have outstanding mysteries and open questions. There are still things to understand at every level of complexity.
The role of precision
All that said, I'm fairly sure that if my younger self had heard the above arguments, some part of him would remain unconvinced. The overriding cause for my mistaken conception of knowledge wasn't so much about the history of different fields or the validity of reductionism. My thinking ultimately rested on a theory about meaning and language.
The question I imagine my younger self asking is: can you really understand something if you don't understand its parts fully? Isn't it, on some level, meaningless to invoke concepts and definitions from fields you haven't studied?
The underlying assumption is that meaning only comes from perfect precision. That there is some way to start at the "bottom"—not having to assume anything or take any concept for granted—and work your way up meticulously to a theory of reality, using perfectly unambiguous terms all along the way.
But perfect precision in language is impossible. Precision is important, but you only ever need enough precision for the problem you’re trying to solve. You need just enough precision to formulate your ideas in a way that can be understood by others (and by your future self), so that you can criticize each other's theories and make progress. In other words, the goal is clarity more so than precision.
In the years since college, I've become convinced that there's no one field that serves as a "foundation" for all our other knowledge. Per the philosopher Karl Popper:
From the point of view of what we want as scientists—understanding, prediction, analysis, and so on—the world in which we live is extremely complex. I should be tempted to say that it is infinitely complex, if the phrase had any meaning. We do not know where or how to start our analysis of this world. There is no wisdom to tell us. Even the scientific tradition does not tell us. It only tells us where and how other people started and where they got to.
The only “foundation” for our current knowledge is whatever our predecessors have learned from testing out the ideas of their time. All of our theories are open to revision, and they’re continually being revised and reworked as we understand the world better. And very rarely do the ideas in one field logically determine the ideas in a different field. We could come up with a new theory of physics without having to overthrow all of our knowledge of biology.2 Same with any revisions that happen in math, logic, philosophy, and other fields.
Rather than thinking of some fields as foundations for others, I now think of some fields as tools for better understanding other fields. Epistemology doesn’t underpin all our other knowledge, but it does give us a framework for understanding how knowledge grows. Mathematics does not underpin physics, but is (among other things) something we can use to formulate theories in physics. All of these entities and fields exist, and they are in large part autonomous—we can study any one of them without having completely solved the others.3
First principles thinking
Is all of this somehow a rejection of the “first principles” thinking that we started with? Far from it. Thinking from first principles is powerful and effective. The lesson I've learned is just that there's no way to do this at a global level—there's no way to trace back to the “bottom” of all our knowledge, because our knowledge does not start at any one prescribed point. First principles thinking is great locally, and intractable globally. Popper:
Science does not rest upon solid bedrock. The bold structure of its theories rises, as it were, above a swamp. It is like a building erected on piles. The piles are driven down from above into the swamp, but not down to any natural or ‘given’ base; and if we stop driving the piles deeper, it is not because we have reached firm ground. We simply stop when we are satisfied that the piles are firm enough to carry the structure, at least for the time being.
The reason I add the caveat "when it comes to understanding the relationship between different fields" is that I'd like to reject methodological reductionism (which pertains to how we should go about understanding the world) while still being agnostic about ontological reductionism (which pertains to what really exists).
In fact, this has already happened—we didn't need to revise much about evolution or cell biology when we discovered special relativity and quantum mechanics! We completely upended our understanding of physics, but did not have to invalidate our understanding of cells and brains.
In the course of talking about the relationships between different fields, I don’t want to overemphasize or reify the boundaries between one field and another. Like all our other terms and ideas, the concepts of “math” and “physics” and “astronomy” are imperfect and their meanings will evolve over time. Ultimately the distinctions between subjects are much less important than the content within them.